I am looking for an example of a first-order sentence in the signature of the real numbers, $(+,\times, <, 0,1)$, that is true when translated in the language of set theory in the natural way, but can be forced to be false.
I'm noticing that many of the standard examples of statements about the reals that are not preserved by forcing, like the continuum hypothesis, are second-order. So I am wondering if there are any first-order examples.
No, there aren’t any first-order examples. The first order theory of the reals in this signature is decidable, so its truth values are determined by arithmetic, which is invariant under forcing. (More to the point, effectively decidable things are just straightforwardly decided by ZFC... that's one way we can conceive of 'deciding' them.)
This answer is a footnote to spaceisdarkgreen's answer, responding to a comment by the OP:
If we add a unary predicate naming the naturals (or integers or similar) we wind up, perhaps surprisingly, with a hugely powerful structure. Specifically, since the naturals support a definable pairing operation, we can now talk about binary relations coded by reals. (For a more technical example of how this plays out, see this old MO question of mine.) This ultimately gives us the full second-order theory of the natural numbers. For example, the $\Sigma^1_3$ sentence "There is a nonconstructible set of naturals" is now appropriately translatable into the theory of this structure, so if $V=L$ we do indeed get that $Th(\mathbb{R};+,\times,\mathbb{N})$ can be altered by forcing.
Interestingly, though, this is not quite the end of the story! In ZFC alone we have Shoenfield absoluteness which says that $\Pi^1_2$ sentences can't have their truth values changed by forcing. While (as the example in the above paragraph shows) this is the best we can do in ZFC alone, in the presence of large cardinals we can say quite a bit more. In particular, given enough large cardinal structure - infinitely many Woodin cardinals, if I recall correctly - the second-order theory of the naturals becomes forcing-invariant. Absoluteness modulo large cardinals even persists into the lowest levels of third-order arithmetic, although the necessary contingency (:P) of CH puts a hard limit on what we can do.
